Application of the Reciprocity Theorem to Analyze Ultrasound Generated by High-Intensity Surface Heating of Elastic Bodies

Abstract

The global reciprocity theorem for thermo-anisotropic elastodynamics, which relates body forces, surface tractions, displacements, temperatures and strains of two solutions of the field equations, states A and B, by integrals over a volume V and its bounding surface S, is derived and discussed in this article. The difference between various anisotropies enters via the stress-temperature tensor, which is different for different cases of anisotropy. The reciprocity theorem is used to analyze the dynamic response to high-intensity heating of a small surface region. The particular example of line-focused laser-beam irradiation of one side of a homogeneous elastic plate is worked out in some detail. For the time-harmonic case, the amplitude of the lowest anti-symmetric mode is obtained in a simple and elegant manner by application of the reciprocity theorem, where the desired mode is state A and state B is taken as a free anti-symmetric mode. The amplitude of any anti-symmetric or symmetric mode can be obtained in an analogous manner, and an appropriate selection of modes can be used with Fourier superposition to obtain the transient response.